Matching wavelet packets to Gaussian random processes

In this paper, we consider the problem of approximating a set of arbitrary, discrete-time, Gaussian random processes by a single, representative wavelet-based, Gaussian process. We measure the similarity between the original processes and the wavelet-based process with the Bhattacharyya coefficient. By manipulating the Bhattacharyya coefficient, we reduce the task of defining the representative process to finding an optimal unitary matrix of wavelet-based eigenvectors, an associated diagonal matrix of eigenvalues, and a mean vector. The matching algorithm we derive maximizes the nonadditive Bhattacharyya coefficient in three steps: a migration algorithm that determines the best basis by searching through a wavelet packet tree for the optimal unitary matrix of wavelet-based eigenvectors; and two separate fixed-point algorithms that derive an appropriate set of eigenvalues and a mean vector. We illustrate the method with two different classes of processes: first-order Markov and bandlimited. The technique is also applied to the problem of robust terrain classification in polarimetric SAR images.